# Urysohn equation

A non-linear integral equation of the form

$$\tag{* } \phi ( x) = \lambda \int\limits _ \Omega K ( x , s , \phi ( s) ) d s + f ( x) ,\ \ x \in \Omega ,$$

where $\Omega$ is a bounded closed set in a finite-dimensional Euclidean space and $K ( x , s , t )$ and $f ( x)$ are given functions for $x , s \in \Omega$, $- \infty < t < \infty$. Suppose that $K ( x , s , t )$ is continuous for the set of variables $x , s \in \Omega$, $| t | \leq \rho$( where $\rho$ is some positive number), and let

$$\left | \frac{\partial K ( x , s , t ) }{\partial t } \right | \leq M = \textrm{ const } ,\ \ x , s \in \Omega ,\ \ | t | \leq \rho .$$

If

$$| \lambda | M \mathop{\rm meas} ( \Omega ) < 1 ,$$

$$| \lambda | \max _ {x \in \Omega } \int\limits _ \Omega \max _ {| t | \leq \rho } | K ( x , s , t ) | d s \leq \rho ,$$

then the equation

$$\phi ( x) = \lambda \int\limits _ \Omega K ( x , s , \phi ( s) ) d s$$

has a unique continuous solution $\phi ( x)$, $x \in \Omega$, satisfying the inequality $| \phi ( x) | \leq \rho$. If $\phi _ {0}$ is any continuous function satisfying $| \phi _ {0} ( x) | \leq \rho$( $x \in \Omega$), then the sequence of approximations

$$\phi _ {n} ( x) = \lambda \int\limits _ \Omega K ( x , s , \phi _ {n-} 1 ( s) ) d s ,\ \ n = 1 , 2 \dots$$

converges uniformly on $\Omega$ to $\phi ( x)$.

Let the Urysohn operator

$$A \phi ( x) = \ \int\limits _ \Omega K ( x , s , \phi ( s) ) d s$$

act in the space $L _ {p} ( \Omega )$, $p > 1$, and let for all $t _ {1} , t _ {2}$, $x , s \in \Omega$ the inequality

$$| K ( x , s , t _ {1} ) - K ( x , s , t _ {2} ) | \leq K _ {1} ( x , s ) | t _ {1} - t _ {2} |$$

be fulfilled, where $K _ {1}$ is a measurable function satisfying

$$\Delta ^ {p} = \ \int\limits _ \Omega \left ( \int\limits _ \Omega K _ {1} ^ {p / ( p - 1 ) } ( x , s ) d s \right ) ^ {p-} 1 d x < \infty .$$

Then for $| \lambda | < \Delta ^ {-} 1$ and $f \in L _ {p} ( \Omega )$, equation (*) has a unique solution in $L _ {p} ( \Omega )$.

Under certain assumptions, equation (*) was first studied by P.S. Urysohn (cf. Non-linear integral equation).

How to Cite This Entry:
Urysohn equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Urysohn_equation&oldid=49100
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article